Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.55·2-s + 3-s + 4.51·4-s − 1.95·5-s − 2.55·6-s − 6.42·8-s + 9-s + 4.97·10-s + 5.66·11-s + 4.51·12-s − 4.68·13-s − 1.95·15-s + 7.36·16-s − 4.34·17-s − 2.55·18-s + 5.53·19-s − 8.80·20-s − 14.4·22-s − 3.74·23-s − 6.42·24-s − 1.19·25-s + 11.9·26-s + 27-s − 0.177·29-s + 4.97·30-s − 0.750·31-s − 5.96·32-s + ⋯
L(s)  = 1  − 1.80·2-s + 0.577·3-s + 2.25·4-s − 0.872·5-s − 1.04·6-s − 2.27·8-s + 0.333·9-s + 1.57·10-s + 1.70·11-s + 1.30·12-s − 1.29·13-s − 0.503·15-s + 1.84·16-s − 1.05·17-s − 0.601·18-s + 1.27·19-s − 1.96·20-s − 3.08·22-s − 0.781·23-s − 1.31·24-s − 0.239·25-s + 2.34·26-s + 0.192·27-s − 0.0329·29-s + 0.908·30-s − 0.134·31-s − 1.05·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 2.55T + 2T^{2} \)
5 \( 1 + 1.95T + 5T^{2} \)
11 \( 1 - 5.66T + 11T^{2} \)
13 \( 1 + 4.68T + 13T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
19 \( 1 - 5.53T + 19T^{2} \)
23 \( 1 + 3.74T + 23T^{2} \)
29 \( 1 + 0.177T + 29T^{2} \)
31 \( 1 + 0.750T + 31T^{2} \)
37 \( 1 - 8.28T + 37T^{2} \)
43 \( 1 - 6.12T + 43T^{2} \)
47 \( 1 + 0.738T + 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 - 4.83T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 - 9.26T + 73T^{2} \)
79 \( 1 - 8.55T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 6.24T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.82618524533092499900691662753, −7.33382672153668520652336772763, −6.77414244199097937583936350249, −6.01845895506967371791736954465, −4.60896582518648086913560954547, −3.88394075749080424534448401242, −2.91004949399538655597404175550, −2.05153728292498832648727417824, −1.14332759804562915405097110772, 0, 1.14332759804562915405097110772, 2.05153728292498832648727417824, 2.91004949399538655597404175550, 3.88394075749080424534448401242, 4.60896582518648086913560954547, 6.01845895506967371791736954465, 6.77414244199097937583936350249, 7.33382672153668520652336772763, 7.82618524533092499900691662753

Graph of the $Z$-function along the critical line